3. 4. 3 Laplace Operator

In the special case of the Laplace operator the vector $ \mathbf{d}$ has the following form

$\displaystyle \mathbf{d} = [0, 0, 1, 0, 1, 0, \ldots]$ (3.60)

For a grid with a regular distance $ h$ between neighboring vertices, the geometrical coefficient matrix $ \mathcal{G}$ yields for five and nine neighboring points:

$\displaystyle \mathcal{G}_5 =
\left[
\begin{array}{c c c c c}
1 & 0 & 0 & 0 & 0...
...2 & 0 \\
1 & 0 & h & 0 & h^2/2 \\
1 & 0 & -h & 0 & h^2/2
\end{array}\right]
$

\begin{displaymath}
\mathcal{G}_9 =
\left[
\begin{array}{c c c c c c c c c}
1& ...
... &-2h& 0 & h^2 & 0 & -3h^3/2 & 0 & 8h^4/3
\end{array}\right]
\end{displaymath}

After eliminating derivatives which do not appear in the series expansion of the single points, the matrices can be re-written. The derivative vector $ [\partial]$ is written as:

$\displaystyle [\partial] = [u, \partial_x u, \partial_y u, \partial_{xx} u, \pa...
...tial_xxx u, \partial_{yyy} u, \partial_{xxxx} u, \partial_{yyyy} u \ldots] \; .$ (3.61)

The vector $ \mathbf{d}$ is reduced to the following form:

$\displaystyle \mathbf{d} = [0, 0, 1, 1, 0, \ldots]$ (3.62)

Inserting into the formula (3.58) yields the well known expressions. The function $ K(\mathbf{v}, \mathbf{w})$ can be written as

\begin{displaymath}
K(\mathbf{w}, \mathbf{v}) := \left\{
\begin{array}{c\vert ...
...} \\
\mathbf{v} \neq \mathbf{w} & h^{-2}
\end{array} \right.
\end{displaymath}

For nine points the following formula is obtained:

\begin{displaymath}
K(\mathbf{w}, \mathbf{v}) := \left\{
\begin{array}{c \vert...
...\mathbf{w}, \mathbf{v}) = 2h & -1/12h^{-2}
\end{array} \right.
\end{displaymath}

Michael 2008-01-16